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12. Sine and the Area of a Triangle
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12. 

Sine and the Area of a Triangle

In geometry, the area of a triangle is traditionally calculated using its base and height. However, there's a trigonometric approach that offers a fresh perspective. By knowing the lengths of two sides of a triangle and the measure of their included angle, one can determine the triangle's area using the sine function. This method is especially useful when the height of the triangle is not directly provided or when it's challenging to determine. The sine-based formula provides a versatile tool for calculating areas, especially in complex geometric scenarios. For instance, considering real-world applications, this approach can be used to determine areas of irregular plots or regions, enhancing our understanding and application of geometry in practical situations.
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10 Theory slides
11 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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Sine and the Area of a Triangle
Slide of 10
The area of a triangle is half the product of its base and the corresponding height. In this lesson, a formula for the area of a triangle involving the sine ratio will be derived.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Investigating the Area of a Triangular Table

Izabella has a triangular table in her kitchen. She wants to cover the surface of the table with a tablecloth. Two side lengths and two angle measures of the table are given.

Triangular Table
How much fabric does Izabella need to cover the surface of the table?
Explore

Investigating the Area of a Triangle

Typically to find the area of a triangle, the measurements of its height and base are used. On the other hand, if the lengths of two sides and the measure of their included angle are known, what should be the process to find the area of the triangle?

In the lengths of and are and inches, respectively. The measure of varies from to
What information is needed to find the area of How can the measure of be used to find the area of
Example

Area of an Equilateral Triangle

One way of finding the area of an equilateral triangle is by using trigonometric ratios.

Izabella wants to finish her math homework before she goes out to buy some fabric for her table. She is given an equilateral triangle whose side length is unit.

Equilateral Triangle

Izabella's homework problem is to find the area of this triangle. Help her find the answer.

Hint

What information is needed to find the area of a triangle?

Solution

The area of a triangle is half the product of its base and the corresponding height. Therefore, the height of needs to be found. Suppose line bisects and intersects at point

Since is an equilateral triangle, the measure of each of its interior angles is Consequently, measures

Considering it can be concluded that is a right angle by the Triangle Angle Sum Theorem.

Consequently, is an altitude of and its corresponding base. To find the right triangle whose vertices are and will be considered. Note that the hypotenuse of is and that is the opposite side to the angle whose measure is Therefore, the sine ratio can be used to find
Recall that the exact value of is
Now that the height was found, the area of can be calculated.
SubstituteII

,

▼
Multiply
IdPropMult

Identity Property of Multiplication

MultFrac

Multiply fractions

The area of the equilateral triangle is
Explore

Investigating the Area of Obtuse Triangles

In the previous example, it was shown that trigonometric ratios can be used to find the area of a triangle given the lengths of two sides and the measure of their included acute angle. What happens if the included angle is obtuse?

In the lengths of and are and inches, respectively. The measure of varies from to
How can the measure of be used to find the area of when is obtuse? What is the relation between the areas of when the measure of is degrees and degrees? What is the relation between the sine ratio of an acute angle and its supplementary angle?
Example

Finding the Area of an Obtuse Triangle

Izabella finished her homework and is ready to go. On her way out, she notices an empty triangular region in the garden.

Her father wants to fill this region with topsoil. To figure out how much topsoil he needs, he asks Izabella to find the area of the region. Help them find the area. If it is necessary, round the answer to the nearest square foot.

Hint

Begin by drawing the external altitude of the side.

Solution

Begin by drawing the external altitude of the side.

Notice that the angle opposite the altitude is the supplementary angle of Its measure can be found by subtracting from
The triangle formed by the altitude, the side whose length is feet, and the extension of the base, is a right triangle. Here, the hypotenuse is feet and the measure of the opposite angle to the altitude is Therefore, to find the height the sine ratio can be used.
The above equation can be solved for
▼
Solve for
MultEqn

RearrangeEqn

Rearrange equation

UseCalc

Use a calculator

Now that the height was found, the formula for the area of a triangle can be used to find the area of the triangular region.
SubstituteII

,

▼
Evaluate right-hand side
MoveRightFacToNumOne

CalcQuot

Calculate quotient

UseCalc

Use a calculator

RoundInt

Round to nearest integer

The area of the region is about square feet.
Discussion

Area of a Triangle Using Sine

There is another way of finding the area of a triangle that can be derived from the previous examples.

The area of a triangle is half the product of the lengths of any two sides and the sine of their included angle. There are three possible formulas for every triangle.
Triangle
The proof of this theorem will be developed using the triangle shown, where is obtuse. However, the same proof is valid for all triangles.

Proof

To find the first formula, start by drawing the altitude from and let be its length. Since the altitude is perpendicular to the base, it generates two right triangles.

Triangle with one altitude drawn
Because is a right triangle, the height of the triangle can be related to the sine of using the sine ratio.
Next, substitute the expression found for into the general formula for the area of a triangle.
The first formula was obtained. To obtain the second formula, notice that is also a right triangle. Therefore, the sine ratio can be applied again, this time to connect and
By substituting this expression into the general formula for the area of a triangle, the second formula can be obtained.

To deduce the third formula, the altitude from or should be drawn. In this case, the altitude from will be arbitrarily drawn and labeled with a length of Because is obtuse, the altitude will lie outside the triangle.

In this case, the length of the base is and the height is Since is a right triangle, the sine ratio can be used to connect and
Since and form a linear pair, they are supplementary angles. Recall that the sine of an angle is equal to the sine of its supplementary angle. With this information, and using the Substitution Property of Equality, a formula connecting and can be written.
Multiplying both sides of the last equation by it is obtained that Finally, substitute this expression for into the formula for the area of
Therefore, for any triangle, its area can be found by calculating half the product of the length of two of its sides and the sine of their included angle.
Example

Using Sine to Calculate Areas in the Real World

While sitting on the bus on the way to the fabric store, Izabella began daydreaming about a formula for the area of a triangle involving the sine ratio. She dreams about applying this formula to find the area of the Bermuda Triangle which she remembers from the movie Gulliver's Travels.

Given the lengths of two sides and the measure of their included angle, calculate the area of the Bermuda Triangle. Round the answer to the nearest square mile.

Hint

Use the formula for the area of a triangle involving the sine ratio.

Solution
Recall the formula for the area of a triangle involving the sine ratio.
Since the lengths of two sides and the measure of their included angle are known, and can be substituted into the formula to find the area.
SubstituteValues

Substitute values

▼
Evaluate right-hand side
Multiply

Multiply

MoveRightFacToNumOne

CalcQuot

Calculate quotient

UseCalc

Use a calculator

RoundInt

Round to nearest integer

Therefore, the area of the Bermuda Triangle is approximately square miles.
Pop Quiz

Practice Using Sine Finding the Area of a Triangle

In the following triangles, the lengths of two sides and the measure of their included angle is given. By using the formula derived earlier, find the area of the triangles. Round the answer to one decimal place.

Closure

Finding the Area of a Triangular Table

By using a formula derived in this lesson, the challenge presented at the beginning can be solved. Recall that the question asked what amount of fabric Izabella needs to cover the triangular table.

Triangular Table
Write the answer correct to one decimal place.

Hint

Begin by finding the included angle. Then use the formula for the area of a triangle using the sine ratio.

Solution

To find how much fabric Izabella needs, the surface area of the table must be found. Therefore, begin by finding the included angle.

Triangular Table
To do so, the Triangle Angle Sum Theorem can be used.
The lengths of two sides and the measure of their included angle are known. Therefore, the formula for the area of a triangle including the sine ratio can be used.
▼
Evaluate right-hand side
Multiply

Multiply

MoveRightFacToNumOne

CalcQuot

Calculate quotient

UseCalc

Use a calculator

RoundDec

Round to decimal place(s)

The area of the table is about square feet. Therefore, Izabella needs about square feet of fabric.



Level 1
Level 2
Level 3
Sine and the Area of a Triangle

Determine the area of the triangle in centimeters. Round the answer to one decimal place.

a green triangle where two of the sides are 5.4 cm and 6 cm and they have an included angle of 22 degrees
a blue triangle with sides of 4.6 cm and 5.1 cm and an included angle of 24 degrees
a red triangle with sides of 2.5 and 3.54 cm as well as an included angle of 63 degrees

Examining the diagram, we see that we know two sides and their included angle. Let's recall the formula of Area of a Triangle Using Sine. A=1/2absin C This formula allows for us to calculate the area of a triangle if we know the length of two sides a and b and the included angle C.


For this triangle, we also know two sides and one angle. Therefore, we can use the formula Area of a Triangle Using Sine, once again, to determine the area.


As in Part A and B, we have the necessary information to determine the triangle's area using sine.


Determine the area of the triangle in centimeters. Round the answer to one decimal place.

a triangle with sides of 5.1 and 3.8 cm and two known angles of 38 degrees and 57 degrees
a triangle with sides of 4.3 and 4.1 cm and where the exterior angle of one of the triangles angle is known

In the given triangle, we know two angles. To be able to use the formula for calculating the triangle's area, we need to determine the remaining angle by using the Interior Angles Theorem. m∠ C + 38^(∘) +57^(∘) = 180^(∘) ⇓ m∠ C =85^(∘) Now that we know two sides and their included angle, we can determine the area of the triangle.

The area of the triangle is about 9.7 square centimeters.

Notice that we have been given the exterior angle of one of the angles in the triangle. Since these two angles form a linear pair, we can determine the inner angle by equating these angles with 180^(∘) and solving the resulting equation. m∠ C + 136^(∘) = 180^(∘) ⇓ m∠ C =44^(∘) Now we can calculate the area.

The triangle has an area of about 6.3 square centimeters.

In the following diagrams, the areas and leg lengths are given in centimeters. Determine the measure of the acute angle of the triangle. Round the answer to the nearest whole number of degrees.

a red triangle with an area of 4.94 square centimeters and where two of the sides are 2.6 and 4.1 centimeters
a triangle with area of 2.22 square centimeters and two sides of 5.2 and 2.5 centimeters

By substituting the sides and the given area into the formula for calculating the Area of a Triangle Using Sine, we can solve for the included angle.

As we can see, the measure of the acute ∠ v is about 68^(∘).

We will solve for ∠ v in the same way as we did in Part A.

In this triangle, the measure of the acute angle labeled v is about 20^(∘).

An equilateral triangle has sides with lengths of centimeters. What is its area? Round the answer to one decimal place.

An equilateral triangle is also equiangular. This means that it has three congruent angles, each with a measure of 60^(∘). Let's draw the triangle.

Let's determine the triangle's area by using the formula for the Area of a Triangle Using Sine.

The area of the triangle is about 15.6 square centimeters.

What is the area of the quadrilateral in centimeters? Round the answer to one decimal place.

a quadrilateral with sides of 2.8 cm, 4.2 cm, 4.1 cm, and 2.2 cm as well as two opposite angles of 59 degrees and 108 degrees

To determine the area of the quadrilateral, we will divide it into two triangles and label them A_1 and A_2.

Notice that we now have two triangles where we know two sides and their included angle. This means we can calculate the area of the triangles using sine. Let's substitute each set of information into the formula for calculating the area. A_1&=1/2(4.2)(4.1)(sin 59^(∘)) [0.9em] A_2&=1/2(2.8)(2.2)(sin 108^(∘)) The sum of these equals the area of the quadrilateral, which we will label A_Q.

The quadrilateral's area is about 10.3 square centimeters.

Sine and the Area of a Triangle

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